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PRISM Manual | PropertySpecification / TheSOperator 
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    <p><a class='wikilink' href='Introduction.html'>Property Specification</a> /
</p><h1>The S Operator</h1>

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<p>The <code><strong>S</strong></code> operator is used to reason about the <em>steady-state</em> behaviour of a model,
i.e. its behaviour in the <em>long-run</em> or <em>equilibrium</em>.
Although this could in principle relate to all three model types,
PRISM currently only provides support for DTMCs and CTMCs.
The definition of steady-state (long-run) probabilities for finite DTMCS and CTMCs is well defined (see e.g. [<a class='wikilink' href='../Main/References.html#Ste94'>Ste94</a>]).
Informally, the property:
</p>
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  <div class='sourceblocktext'><div class="prism"><span class="prismkeyword">S</span> <span class="prismident">bound</span> [ <span class="prismident">prop</span> ]<br/>
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  <div class='sourceblocklink'><a href='TheSOperator@action=sourceblock&amp;num=1' type='text/plain'>[&#036;[Get Code]]</a></div>
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<p class='vspace'>is true in a state <em>s</em> of a DTMC or CTMC if
"starting from <em>s</em>, the steady-state (long-run) probability of being in a state which satisfies the (Boolean-valued) PRISM property <code>prop</code>, meets the bound <code>bound</code>".
A typical example of this type of property would be:
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  <div class='sourceblocktext'><div class="prism"><span class="prismkeyword">S</span>&lt;<span class="prismnum">0.05</span> [ <span class="prismident">queue_size</span> / <span class="prismident">max_size</span> &gt; <span class="prismnum">0.75</span> ]<br/>
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  <div class='sourceblocklink'><a href='TheSOperator@action=sourceblock&amp;num=2' type='text/plain'>[&#036;[Get Code]]</a></div>
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<p class='vspace'>which means: "the long-run probability of the queue being more than 75% full is less than 0.05".
</p>
<p class='vspace'>Like the <code><strong>P</strong></code> <a class='wikilink' href='ThePOperator.html'>operator</a>, the <code><strong>S</strong></code> operator can be used in a <em>quantitative</em> form, which returns the actual probability value, e.g.:
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  <div class='sourceblocktext'><div class="prism"><span class="prismkeyword">S</span>=? [ <span class="prismident">queue_size</span> / <span class="prismident">max_size</span> &gt; <span class="prismnum">0.75</span> ]<br/>
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  <div class='sourceblocklink'><a href='TheSOperator@action=sourceblock&amp;num=3' type='text/plain'>[&#036;[Get Code]]</a></div>
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<h3><a class='wikilink' href='../Main/Welcome.html'>PRISM Manual</a></h3>
<p><strong><a class='wikilink' href='Introduction.html'>Property Specification</a></strong>
</p><ul><li><a class='wikilink' href='Introduction.html'>Introduction</a>
</li><li><a class='wikilink' href='IdentifyingASetOfStates.html'>Identifying A Set Of States</a>
</li><li><a class='wikilink' href='ThePOperator.html'>The P Operator</a>
</li><li><a class='selflink' href='TheSOperator.html'>The S Operator</a>
</li><li><a class='wikilink' href='Reward-basedProperties.html'>Reward-based Properties</a>
</li><li><a class='wikilink' href='SyntaxAndSemantics.html'>Syntax And Semantics</a>
</li><li><a class='wikilink' href='Filters.html'>Filters</a>
</li><li><a class='wikilink' href='PTAProperties.html'>PTA Properties</a>
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